Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz gave an extensional definition of the modal operator: $\Diamond A =_{def} \neg A \to A$ (which he attributes to Tarski).

This gives a weird modal logic, with some paradoxical, if not seemingly absurd theorems, notably $(\Diamond A \land \Diamond B) \to \Diamond(A\land B)$. Substitute $\neg A$ for $B$ to see why it's been relegated to a footnote in the history of modal logic.

Anyhow, the only non-critical work that I found a reference to is a talk by A. Turquette, "A generalization of Tarski's Möglichkeit" at the Australasian Association for Logic 1997 Annual Conference. The abstract is in the BSL 4 (4), http://www.math.ucla.edu/~asl/bsl/0404/0404-006.ps Basically Turquette suggested applications in m-valued logics for m-state systems. (I've not been able to obtain any notes, slides or other content of this talk, so I would appreciate hearing from anyone who has more information.)

I don't have any applications for it, but I find the properties to be interesting enough to merit a paper (in progress) on adding this operator to various substructural logics, and comparing those logics with themselves augmented by Lewsian modal operators.

My question: Is anyone here aware of other articles or papers on Tarski's Möglichkeit or "extensional" modalities?

I have no references for you. But another context in which the seemingly absurd theorem would be reasonable is if $\square A$ were intended to mean "$A$ is provable intuitionistically." So perhaps there are connections to be had with intuitionistic provability logic (of which I know very little)?
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Ed DeanApr 13 '11 at 17:46

@Ed $\Box$ doesn't seem to coincide with intuitionistic provability, though I am not familiar w/intuitionistic provability logic. Do you have references? One result I have is that for modal theorems proved in MALL+K, the corresponding Tarskian-modal theorems are proved in MALL. (Though MALL can prove some modal theorems not provable in corresponding MALL+K.) A preliminary paper is online at arxiv.org/abs/1105.0354 and will be updated.
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RobMay 3 '11 at 13:25

1 Answer
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Rob, I didn't know this was called the Tarskian Möglichkeit, but Martin Escardo and I have been studying this operator (A -> B) -> A, in the more general case when falsity is an arbitrary formula B, for the past few years, mainly in connection with computational interpretations of classical theorems. If we let B be fixed, then we define

J A = (A -> B) -> A

It is easy to show that this is a strong monad. We call it the "selection monad" or the "Peirce monad", as J A -> A is Peirce's law. In fact, the seemingly absurd theorem you mentioned in your post is the cornerstone for our work on interpreting ineffective principles such as the Tychonoff theorem, for instance. Have a look at some of our papers, e.g.

Martín Escardó and Paulo Oliva. Sequential games and optimal strategies. Proceedings of the Royal Society A, 467:1519-1545, 2011.

Any paper which mentions "selection functions" or "game" is related to the operator you are asking about.

I must warn we have been studying this operator in the setting of intuitonistic (minimal) logic. But I find it very interesting that you are looking at this in the more refined (substructural) settings of linear logic and Lukasiewicz logic.